Problem: Let $g$ be a vector-valued function defined by $g(t)=\left(-t^4+2t+5,5\cdot3^{t}\right)$. Find $g$ 's second derivative $g''(t)$. Choose 1 answer: Choose 1 answer: (Choice A) A $\left(12t^2,45\cdot3^t\right)$ (Choice B) B $\left(-t^2,5\ln(3)\cdot3^t\right)$ (Choice C) C $\left(-4t^3+2,5\ln(3)\cdot3^t\right)$ (Choice D) D $\left(-12t^2,5(\ln(3))^2\cdot3^t\right)$
Solution: We are asked to find the second derivative of $g$. This means we need to differentiate $g$ twice. In other words, we differentiate $g$ once to find $g'$, and then differentiate $g'$ (which is a vector-valued function as well) to find $g''$. Recall that $g(t)=(-t^4+2t+5,5\cdot3^{t})$. Therefore, $g'(t)=(-4t^3+2,5\ln(3)\cdot3^t)$. Now let's differentiate $g'(t)=(-4t^3+2,5\ln(3)\cdot3^t)$ to find $g''$. $g''(t)=(-12t^2,5(\ln(3))^2\cdot3^t)$ In conclusion, $g''(t)=\left(-12t^2,5(\ln(3))^2\cdot3^t\right)$.